This is a survey of some problems in geometric group theory which I find
interesting. The problems are from different areas of group theory. Each
section is devoted to problems in one area. It contains an introduction where I
give some necessary definitions and motivations, problems and some discussions
of them. For each problem, I try to mention the author. If the author is not
given, the problem, to the best of my knowledge, was formulated by me first.; Comment: 25 pages

This paper is a guide for the pure mathematician who would like to know more
about cryptography based on group theory. The paper gives a brief overview of
the subject, and provides pointers to good textbooks, key research papers and
recent survey papers in the area.; Comment: 25 pages References updated, and a few extra references added. Minor
typographical changes. To appear in Proceedings of Groups St Andrews 2009 in
Bath, UK

Color-ordered amplitudes for the scattering of n particles in the adjoint
representation of SU(N) gauge theory satisfy constraints that arise from group
theory alone. These constraints break into subsets associated with irreducible
representations of the symmetric group S_n, which allows them to be presented
in a compact and natural way. Using an iterative approach, we derive the
constraints for six-point amplitudes at all loop orders, extending earlier
results for n=4 and n=5. We then decompose the four-, five-, and six-point
group-theory constraints into their irreducible S_n subspaces. We comment
briefly on higher-point two-loop amplitudes.; Comment: 35 pages; v2: typos in eqs. 4.14 and 4.20 corrected; v3: typo in eq.
7.3 corrected

We introduce the classical theory of the interplay between group theory and
topology into the context of operads and explore some applications to homotopy
theory. We first propose a notion of a group operad and then develop a theory
of group operads, extending the classical theories of groups, spaces with
actions of groups, covering spaces and classifying spaces of groups. In
particular, the fundamental groups of a topological operad is naturally a group
operad and its higher homotopy groups are naturally operads with actions of its
fundamental groups operad, and a topological $K(\pi,1)$ operad is characterized
by and can be reconstructed from its fundamental groups operad. Two most
important examples of group operads are the symmetric groups operad and the
braid groups operad which provide group models for $\Omega^{\infty}
\Sigma^{\infty} X$ (due to Barratt and Eccles) and $\Omega^2 \Sigma^2 X$ (due
to Fiedorowicz) respectively. We combine the two models together to produce a
free group model for the canonical stabilization $\Omega^2 \Sigma^2 X
\hookrightarrow \Omega^{\infty} \Sigma^{\infty} X$, in particular a free group
model for its homotopy fibre.; Comment: submitted; 39 pages; part of the author's Ph.D. thesis; Abstract and
Introduction rewritten; Remarks 2.14 and 2.32 added concerning extending any
group and G-space to a group operad and G-operad; Acknowledgements added;
numerous minor corrections and changes made

The point of view of these notes on the topic is to bring out the flavor that
Representation Theory is an extension of the first course on Group Theory. We
also emphasize the importance of base field. These notes cover completely the
theory over complex numbers which is Character Theory. A large number of worked
out examples are the main feature of these notes. The prerequisite for this
note is basic group theory and linear algebra.; Comment: These notes grew out of a course on Representation Theory of finite
groups given to undergraduate students at IISER Pune. This is first draft and
hence readers are advised to use their own judgment

Let $\lambda=(\lambda_1,\lambda_2,...)$ be a \emph{partition} of $n$, a
sequence of positive integers in non-increasing order with sum $n$. Let
$\Omega:=\{1,...,n\}$. An ordered partition $P=(A_1,A_2,...)$ of $\Omega$ has
\emph{type} $\lambda$ if $|A_i|=\lambda_i$.
Following Martin and Sagan, we say that $G$ is \emph{$\lambda$-transitive}
if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of
$\Omega$ of type $\lambda$, there exists $g\in G$ with $A_ig=B_i$ for all $i$.
A group $G$ is said to be \emph{$\lambda$-homogeneous} if, given two ordered
partitions $P$ and $Q$ as above, inducing the sets $P'=\{A_1,A_2,...\}$ and
$Q'=\{B_1,B_2,...\}$, there exists $g\in G$ such that $P'g=Q'$. Clearly a
$\lambda$-transitive group is $\lambda$-homogeneous.
The first goal of this paper is to classify the $\lambda$-homogeneous groups.
The second goal is to apply this classification to a problem in semigroup
theory.
Let $\trans$ and $\sym$ denote the transformation monoid and the symmetric
group on $\Omega$, respectively. Fix a group $H\leq \sym$. Given a
non-invertible transformation $a\in \trans\setminus \sym$ and a group $G\leq
\sym$, we say that $(a,G)$ is an \emph{$H$-pair} if the semigroups generated by
$\{a\}\cup H$ and $\{a\}\cup G$ contain the same non-units...

We give a survey of various recent developments in orbit equivalence and
measured group theory. This subject aims at studying infinite countable groups
through their measure preserving actions.; Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International
Congress of Mathematicians, Hyderabad, India - International Congress of
Mathematicians (ICM), Hyderabad : India (2010)

After some excitement generated by recently suggested public key exchange
protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent
opinion now that the conjugacy search problem is unlikely to provide sufficient
level of security if a braid group is used as the platform. In this paper we
address the following questions: (1) whether choosing a different group, or a
class of groups, can remedy the situation; (2) whether some other "hard"
problem from combinatorial group theory can be used, instead of the conjugacy
search problem, in a public key exchange protocol. Another question that we
address here, although somewhat vague, is likely to become a focus of the
future research in public key cryptography based on symbolic computation: (3)
whether one can efficiently disguise an element of a given group (or a
semigroup) by using defining relations.; Comment: 12 pages

Most common public key cryptosystems and public key exchange protocols
presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve
methods are number theory based and hence depend on the structure of abelian
groups. The strength of computing machinery has made these techniques
theoretically susceptible to attack and hence recently there has been an active
line of research to develop cryptosystems and key exchange protocols using
noncommutative cryptographic platforms. This line of investigation has been
given the broad title of noncommutative algebraic cryptography. This was
initiated by two public key protocols that used the braid groups, one by Ko,
Lee et.al.and one by Anshel, Anshel and Goldfeld. The study of these protocols
and the group theory surrounding them has had a large effect on research in
infinite group theory. In this paper we survey these noncommutative group based
methods and discuss several ideas in abstract infinite group theory that have
arisen from them. We then present a set of open problems.

This is a collection of open problems in Group Theory proposed by more than
300 mathematicians from all over the world. It has been published every 2-4
years in Novosibirsk since 1965, now also in English. This is the 18th edition,
which contains 120 new problems and a number of comments on about 1000 problems
from the previous editions.; Comment: several new solutions and references have been added, as well as a
few corrections; in particular, the corrected version of 2.2 in Archive

Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles. We observe
that there exist Apollonian packings which have strong integrality properties,
in which all circles in the packing have integer curvatures and rational
centers such that (curvature)$\times$(center) is an integer vector. This series
of papers explain such properties. A {\em Descartes configuration} is a set of
four mutually tangent circles with disjoint interiors. We describe the space of
all Descartes configurations using a coordinate system $\sM_\DD$ consisting of
those $4 \times 4$ real matrices $\bW$ with $\bW^T \bQ_{D} \bW = \bQ_{W}$ where
$\bQ_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+
x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and $\bQ_W$ of the quadratic form
$Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the
parameter space $\sM_\DD$. We observe that the Descartes configurations in each
Apollonian packing form an orbit under a certain finitely generated discrete
group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer
matrices, and its integrality properties lead to the integrality properties
observed in some Apollonian circle packings. We introduce two more related
finitely generated groups...

Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles. Such
packings can be described in terms of the Descartes configurations they
contain. It observed there exist infinitely many types of integral Apollonian
packings in which all circles had integer curvatures, with the integral
structure being related to the integral nature of the Apollonian group. Here we
consider the action of a larger discrete group, the super-Apollonian group,
also having an integral structure, whose orbits describe the Descartes
quadruples of a geometric object we call a super-packing. The circles in a
super-packing never cross each other but are nested to an arbitrary depth.
Certain Apollonian packings and super-packings are strongly integral in the
sense that the curvatures of all circles are integral and the
curvature$\times$centers of all circles are integral. We show that (up to
scale) there are exactly 8 different (geometric) strongly integral
super-packings, and that each contains a copy of every integral Apollonian
circle packing (also up to scale). We show that the super-Apollonian group has
finite volume in the group of all automorphisms of the parameter space of
Descartes configurations...

This paper gives $n$-dimensional analogues of the Apollonian circle packings
in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional
oriented Descartes configurations parametrized in a coordinate system,
ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices $\bW$ with $\bW^T
\bQ_{D,n} \bW = \bQ_{W,n}$ where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 -
\frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic
form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and $\bQ_{D,n}$ and
$\bQ_{W,n}$ are their corresponding symmetric matrices. There are natural
actions on the parameter space $\sM_{\dd}^n$. We introduce $n$-dimensional
analogues of the Apollonian group, the dual Apollonian group and the
super-Apollonian group. These are finitely generated groups with the following
integrality properties: the dual Apollonian group consists of integral matrices
in all dimensions, while the other two consist of rational matrices, with
denominators having prime divisors drawn from a finite set $S$ depending on the
dimension. We show that the the Apollonian group and the dual Apollonian group
are finitely presented, and are Coxeter groups. We define an Apollonian cluster
ensemble to be any orbit under the Apollonian group...

Decision problems are problems of the following nature: given a property P
and an object O, find out whether or not the object O has the property P. On
the other hand, witness problems are: given a property P and an object O with
the property P, find a proof of the fact that O indeed has the property P. On
the third hand(?!), search problems are of the following nature: given a
property P and an object O with the property P, find something "material"
establishing the property P; for example, given two conjugate elements of a
group, find a conjugator. In this survey our focus is on various search
problems in group theory, including the word search problem, the subgroup
membership search problem, the conjugacy search problem, and others.

Let K be a principal ideal domain, G a finite group, and M a KG-module which
as K-module is free of finite rank, and on which $G$ acts faithfully. A
generalized crystallographic group (introduced by the authors in volume 5 of
Journal of Group Theory) is a group $\frak C$ which has a normal subgroup
isomorphic to M with quotient G, such that conjugation in $\frak C$ gives the
same action of G on M that we started with. (When $K=\Bbb Z$, these are just
the classical crystallographic groups.) The K-free rank of M is said to be the
dimension of $\frak C$, the holonomy group of $\frak C$ is G, and $\frak C$ is
called indecomposable if M is an indecomposable KG-module.
Let K be either $\Bbb Z$, or its localization $\Bbb Z_{(p)}$ at the prime p,
or the ring $\Bbb Z_p$ of p-adic integers, and consider indecomposable
torsionfree generalized crystallographic groups whose holonomy group is
noncyclic of order p^2. In Theorem 2, we prove that (for any given p) the
dimensions of these groups are not bounded.
For $K=\Bbb Z$, we show in Theorem 3 that there are infinitely many
non-isomorphic indecomposable torsionfree crystallographic groups with holonomy
group the alternating group of degree 4. In Theorem 1, we look at a cyclic G
whose order |G| satisfies the following condition: for all prime divisors p of
|G|...

This article is a survey article on geometric group theory from the point of
view of a non-expert who likes geometric group theory and uses it in his own
research. The sections are: classical examples, basics about
quasiisometry,properties and invariants of groups invariant under
quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a
hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for
proper actions, measurable group theory, some open problems.; Comment: 28 pages. Following the two detailed referee reports we have improved
the exposition and corrected typos. The paper will appear in the Muenster
Journal for Mathematics

We survey some recent advances in the homotopy theory of classifying spaces,
and homotopical group theory. We focus on the classification of p-compact
groups in terms of root data over the p-adic integers, and discuss some of its
consequences e.g. for finite loop spaces and polynomial cohomology rings.; Comment: To appear in Proceedings of the ICM 2010.

We give a precise definition of ``generic-case complexity'' and show that for
a very large class of finitely generated groups the classical decision problems
of group theory - the word, conjugacy and membership problems - all have
linear-time generic-case complexity. We prove such theorems by using the theory
of random walks on regular graphs.; Comment: Revised version